Singular points in generic two-parameter families of vector fields on 2-manifold

The paper’s main theorem (Section 3.5) states that, in generic C^q (q>5) two-parameter families on a compact 2‑manifold, at any fixed parameter either (i) all singular points are hyperbolic; or (ii) there are exactly two non-hyperbolic points, both of type SN0 or AH0; or (iii) there is exactly one non-hyperbolic point of type in W = SN0 ∪ AH0 ∪ SN1 ∪ AH1 ∪ BT, with finiteness of singular points ensured; the proof invokes multi-jet transversality and codimension counts via semi-algebraic jet-classes (notably Theorem 9 and the centralizer lemmas) . The candidate solution proves the same trichotomy with a different but standard jet-bundle approach: it encodes the degeneracies as Diff(M)-invariant submanifolds of the jet bundle of sections, uses parametric/multi‑jet transversality, and performs dimension counts on B^2×M and B^2×M^{(r)}. Its codimension numbers include the two equations v=0 in the ambient section-jet space, whereas the paper’s codimensions are for truncated phase jets, explaining the apparent discrepancy (e.g., paper: codim SN0 = 1 in truncated jets; model: codim SN0 = 3 in section-jets = 1 + 2 for v=0) . Substantively, both derivations lead to the same exclusions (e.g., sets of total codimension > dim source are avoided), the same allowance of at most two non-hyperbolic points (Theorem 2), and the same finite singular set on compact M (paper: isolated by normal forms; model: also via the structure of the universal zero set) . The paper adds a nontrivial justification of the codimension claims (Theorem 9) using centralizer lemmas, which the model cites as classical; thus the paper supplies missing details behind the model’s assumptions. Therefore both are correct, with different (but compatible) proof frameworks.